17,967 is total number of seats available across all NITs conbined as per latest statistics. 

These seats can vary every year depending on seat matrix and management decisions.

Total seats are then bifurcated as per categories and quotas. 

Cutoff trends shared for reference. 

https://engineering.careers360.com/articles/nits-cutoff-for-top-btech-branches

Hey.

The total number of seats offered by all the National Institutes of Technology ( NITs) among various branches are around 17,868.

You can use the link below to see the seat matrix for each.

https://www.google.co.in/amp/s/engineering.careers360.com/articles/iits-iiits-and-gftis-add-1558-btech-seats-nits-reduce-145/amp

Best of luck!